Personally, I only rehire people from projects that went smoothly, not ones where I had to make the urgent phone call.
Teams that "just work" are highly valued. They clear up my attention for other things.
Which means that everyone is playing the game to not be cut.
But I'm not sure the author of this thread works in such a place. In that case the game is different.
In the case where the "urgent midnight fix" is important, it's necessary to promote the visibility of your (just working) team. If visibility is the game, then be visible.
You know how test-driven-dev was always "write the test first"? In that environment a test is always written before any code.
Well in the "ticket closing" scenario it's important to open a ticket, regardless of how trivial, for every code action taken. For every meeting attended. For every scenario dodged. If tickets are the way to score then write tickets.
If "being a hero" is the valuable thing, then be a hero. Be prepared to champion your team every chance you get. Every time you interact with management stress the emergency you just fixed (before it became an emergency.) Tomorrow do it again with the next thing.
Management needs visibility. Be visible. I know, this seems stupid and beneath you. But that's why they call it a job, not playtime.